#9719130

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,3)(4,7,8,6)(9,10,15,17)(11,16,14,24)(12,18,21,13)(19,20,22,23)(25,26)(27,32)(28,29)(30,38)(31,33)(34,39)(35,37)(36,40);(2,3)(6,7)(9,11,15,14)(10,16,17,24)(12,19,21,22)(13,23,18,20)(25,27,31,36)(26,32,33,40)(28,37,39,30)(29,35,34,38);(1,3,5,2)(4,7,8,6)(10,17)(12,21)(16,24)(19,22)(26,33)(27,36)(28,39)(35,38);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,19)(13,17)(14,22)(15,21)(16,20)(23,24)(25,28)(26,34)(27,37)(29,33)(30,36)(31,39)(32,38)(35,40);(9,13,15,18)(10,12,17,21)(11,20,14,23)(16,22,24,19)(25,29,31,34)(26,28,33,39)(27,38,36,35)(30,40,37,32);(9,14,15,11)(10,16,17,24)(12,22,21,19)(13,23,18,20)(25,30,31,37)(26,35,33,38)(27,28,36,39)(29,32,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9626326);